Why is the derivative of the activation functions in neural networks important?
If what you are asking is what is the intuition for using the derivative in backpropagation learning, instead of an in-depth mathematical explanation:

Recall that the derivative tells you a function's sensitivity to change with respect to a change in its input. A high (absolute) value for the derivative at a certain point means that the function is very steep, and a small change in input may result in a drastic change in its output; conversely, a low absolute value means little change, so not steep at all, with the extreme case that the function is constant when the derivative is zero.

Training a neural network essentially amounts to an optimization problem where one wants to minimize a certain value, in this case the error produced by the network on the given training examples. Backpropagation learning can be viewed as a case of gradient descent (the inverse of hill climbing).

If for a moment we assume that your input is only 2-dimensional (just for illustration, the mathematics of course also work for higher dimensions), you could imagine the error function as a landscape with hills, mountains, valleys, ridges etc. You are standing at a high point and want to get down as far as possible. Gradient descent means that, in discrete steps, you always walk down in the direction that has the steepest slope downwards from where you are currently standing, until you eventually reach a (local) minimum.

In order to determine where that steepest slope is, you need the derivative of the activation function. Basically, you want to sort out how much each unit in your network contributes to an error, and adjust in the direction that contributes the most.

Edit: Regarding constant values for a derivative, in the landscape metaphor it would mean that the gradient is the same no matter where you are, so you'll always go in the same direction and never reach an optimum. However, multi-layer networks with linear activation function are kind of besides the point anyhow when you consider that each cell computes a linear combination of its inputs, which then is again a linear function, so the output of the last layer will ultimately be a linear function of the inputs at the first layer. That is to say, anything you can do with a multi-layer net with linear activation functions, you could also achieve with just a single layer.